What is "unfolding"

n. (context often figuratively English) The process by which something unfolds or becomes revealed. vb. (present participle of unfold English)

n. a developmental process; "the flowering of ante-bellum culture" [syn: flowering]

In mathematics, an unfolding of a function is a certain family of functions.

Let M be a smooth manifold and consider a smooth mapping f : M → R. Let us assume that for given x ∈ M and y ∈ R we have f(x) = y. Let N be a smooth k-dimensional manifold, and consider the family of mappings (parameterised by N) given by F : M × N → R. We say that F is a k-parameter unfolding of f if F(x, 0) = f(x) for all x. In other words the functions f : M → R and F : M × {0} → R are the same: the function f is contained in, or is unfolded by, the family F.

Let f : R → R be given by f(x, y) = x + y. An example of an unfolding of f would be F : R × R → R given by

F((x, y), (a, b, c)) = x + y + ay + by + cy.
As is the case with unfoldings, x and y are called variables and a,  b,  and c are called parameters – since they parameterise the unfolding.

In practice we require that the unfoldings have certain nice properties. In R notice that f is a smooth mapping from M to R and so belongs to the function space C(M, R). As we vary the parameters of the unfolding we get different elements of the function space. Thus, the unfolding induces a function Φ : N → C(M, R). The space diff(M) × diff(R),  where diff(M) denotes the group of diffeomorphisms of M etc., acts on C(M, R). The action is given by (ϕ, ψ) ⋅ f = ψ ∘ f ∘ ϕ. If g lies in the orbit of f under this action then there is a diffeomorphic change of coordinates in M and R which takes g to f (and vice versa). One nice property that we may like to impose is that

$$\operatorname{Im}(\Phi) \pitchfork \operatorname{orb}(f)$$
where "$\pitchfork$" denotes " transverse to". This property ensures that as we vary the unfolding parameters we can predict – by knowing how the orbit foliate C(M, R) – how the resulting functions will vary.

There is an idea of a versal unfolding. Every versal unfolding has the property that $\operatorname{Im}(\Phi) \pitchfork \operatorname{orb}(f)$, but the converse is false. Let x, …, x be local coordinates on M, and let O(x, …, x) denote the ring of smooth functions. We define the Jacobian ideal of f,  denoted by J as follows:

$$J_f := \left\langle \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} \right\rangle.$$

Then a basis for a versal unfolding of f is given by quotient

$$\frac{\mathcal{O}(x_1,\ldots,x_n)}{J_f}$$

This quotient is known as the local algebra of f. The dimension of the local algebra is called the Milnor number of f. The minimum number of unfolding parameters for a versal unfolding is equal to the Milnor number; that is not to say that every unfolding with that many parameters will be versal! Consider the function f(x, y) = x + y. A calculation shows that

$$\frac{\mathcal{O}(x,y)}{\langle 2x, 5y^4 \rangle} = \{y,y^2,y^3\} \ .$$

This means that {y, y, y} give a basis for a versal unfolding, and that

F((x, y), (a, b, c)) = x + y + ay + by + cy

is a versal unfolding. A versal unfolding with the minimum possible number of unfolding parameters is called a miniversal unfolding.

Sometimes unfoldings are called deformations, versal unfoldings are called versal deformations, etc.

An important object associated to an unfolding is its bifurcation set. This set lives in the parameter space of the unfolding, and gives all parameter values for which the resulting function has degenerate singularities.

Unfolding can mean:

• Unfolding (functions)
• Unfolding (geometry)
• Unfolding (DSP implementation)
• Equilibrium unfolding
• Unfolding, mathematical method also referred to as Deconvolution
• Unfolding (music)

Unfolding is a public art work by artist Bernhard Heiliger located at the Lynden Sculpture Garden near Milwaukee, Wisconsin. The sculpture has an abstract form; it is installed on the patio.

In Schenkerian analysis, unfolding (German: Ausfaltung) or compound melody is the implication of more than one melody or line by a single voice through skipping back and forth between the notes of the two melodies. In music cognition, the phenomenon is also known as melodic fission.

Unfolding is a type of prolongation, "by means of the unfolding of intervals horizontally." Though the notes skipped between, those heard, may be considered near the foreground, the dyads, those implied, are in the middle or background. Middleground dyads are "unfolded" in the foreground: "intervals conceptually heard as sounding together are separated in time, unfolded, as it where, into a melodic sequence."

Unfolding is a transformation technique of duplicating the functional blocks to increase the throughput of the DSP program in such a way that preserves its functional behavior at its outputs. Unfolding was first proposed by Keshab K. Parhi and David G. Messerschmitt in 1989. . Unfolding in general program is as known as Loop unrolling

Unfolding has applications in designing high-speed and low-power ASIC architectures. One application is to unfold the program to reveal hidden concurrency so that the program can be scheduled to a smaller iteration period, thus increasing the throughput of the implementation. Another application is parallel processing in word level or bit level. Therefore these transformed circuit could increase the throughput and decrease the power consumption.